MinT - Best Known (127−47, 127, s)-Nets in Base 16

Best Known (127−47, 127, s)-Nets in Base 16

(127−47, 127, 583)-Net over F₁₆ — Constructive and digital

Digital (80, 127, 583)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 29, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]
2. digital (51, 98, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 49, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(127−47, 127, 2556)-Net over F₁₆ — Digital

Digital (80, 127, 2556)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−47, 127, 2482253)-Net in Base 16 — Upper bound on s

There is no (80, 127, 2482254)-net in base 16, because

1 times m-reduction [i] would yield (80, 126, 2482254)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 52 374272 858238 452513 531010 783096 000571 836134 328107 139712 063887 370861 037422 021746 777034 034439 718976 866683 691765 133582 259914 950542 685886 338797 609157 986256 > 16¹²⁶ [i]